Spherically symmetric self-dual Yang-Mills instantons on curved backgrounds in all even dimensions

We present several different classes of self-dual Yang-Mills instantons in all even $d$-dimensional backgrounds with Euclidean signature. In $d=4p+2$ the only solutions we found are on constant curvature dS (de Sitter) and AdS (anti–de Sitter) backgrounds and are evaluated in closed form. In $d=4p$ an interesting class of instantons are given on black hole backgrounds. One class of solutions are (Euclidean) time-independent and spherically symmetric in $d-1$ dimensions, and the other class are spherically symmetric in all $d$ dimensions. Some of the solutions in the former class are evaluated numerically, all the rest being given in closed form. Analytic proofs of existence covering all numerically evaluated solutions are given. All instantons studied have finite action and vanishing energy momentum tensor and do not disturb the geometry.

Figure 1

The YM gauge functions are shown as a function of the radial coordinate $r$ for two type I $p=2$ YM self-dual solutions, showing the deformation of the explicit DC solution with $c_2=0$ by the $c_2=0.01$ 2-Reissner-Nordström metric.

Figure 2

The parameters $w_h$, $w^{\prime }(r_h)$, and $u^{\prime }(r_h)$ of the $p=2$ type II YM instantons in a $d=8$ Schwarzschild-Tangherlini background are plotted as a function of $r_h$.

Figure 3

The YM gauge functions are shown as a function of the radial coordinate $r$ for $p=2$ type II YM self-dual solutions in $d=8$ Euclideanized Reissner-Nordström-Tangherlini backgrounds.